How many runs can a string contain?

Given a string , a repetition of period p in is a substring , , r≥2, where neither nor is a repetition. The maximum number of repetitions in any string is well known to be Θ(nlogn). A run or maximal periodicity of period p in is a substring of , where is a repetition, a proper prefix of , and no repetition of period p begins at position i of or ends at position .In 2000 Kolpakov and Kucherov showed that the maximum number ρ(n) of runs in any string is O(n), but their proof was nonconstructive and provided no specific constant of proportionality. At the same time, they presented experimental data to prompt the conjecture: ρ(n)